Scott Mayers wrote:( I ) Can we find some universal logic that can cover the full range of all specific logical systems ?

( 2 ) Are we permitted to use any logic especially some possible universal one with initial simple premises
( including possibly none ) to prove all of reality beyond mere abstraction such as scientific truths ?

The notion of a universal logic is flawed because universal knowledge applied to any discipline is simply not possible
Knowledge by definition has to be a posteriori but science is inductive so its knowledge base can never be complete
No logical system outside of science could absolutely prove it for this very reason but could do so if it was deductive

Your opinion about these motives are not in question here at this point, only to the fact that these questions were asked and motivated by those who helped shape the history of both philosophy of logic and science. They didn't assume your conclusion prior to investigating it .

a) “Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true,[1] but not provable in the theory (Kleene 1967, p. 250)

note
"... there is an arithmetical statement that is true..."

In other words there are true mathematical statements which cant be proven
But the fact is Godel cant tell us what makes a mathematical statement true thus his theorem is meaningless

b) Gödel uses a G statement to prove his theorem but Godel's G statement is outlawed by the very axiom of the system he uses to prove his theorem -thus his proof is invalid,

Godel's G statement is impredicative '
Godel uses the axiom of reducibility -ie the logic of the system he is using to make his proof
but
the axiom of reducibility bans impredicative statements
thus Godel's G statement is banned-but he uses it to prove his theorem
thus Godel's theorem is invalid

2)Godel's 2nd theorem ends in paradox

Godel's 2nd theorem is about

"If an axiomatic system can be proven to be consistent and complete from
within itself, then it is inconsistent.”

But we have a paradox

Gödel is using a mathematical system
his theorem says a system cant be proven consistent

THUS A PARADOX

Godel must prove that a system cannot be proven to be consistent based upon the premise that the logic he uses must be consistent . If the logic he uses is not consistent then he cannot make a proof that is consistent. So he must assume that his logic is consistent so he can make a proof of the impossibility of proving a system to be consistent. But if his proof is true then he has proved that the logic he
uses to make the proof must be consistent, but his proof proves that
this cannot be done
THUS A PARADOX

a) “Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true,[1] but not provable in the theory (Kleene 1967, p. 250)

note
"... there is an arithmetical statement that is true..."

In other words there are true mathematical statements which cant be proven
But the fact is Godel cant tell us what makes a mathematical statement true thus his theorem is meaningless

b) Gödel uses a G statement to prove his theorem but Godel's G statement is outlawed by the very axiom of the system he uses to prove his theorem -thus his proof is invalid,

Godel's G statement is impredicative '
Godel uses the axiom of reducibility -ie the logic of the system he is using to make his proof
but
the axiom of reducibility bans impredicative statements
thus Godel's G statement is banned-but he uses it to prove his theorem
thus Godel's theorem is invalid

2)Godel's 2nd theorem ends in paradox

Godel's 2nd theorem is about

"If an axiomatic system can be proven to be consistent and complete from
within itself, then it is inconsistent.”

But we have a paradox

Gödel is using a mathematical system
his theorem says a system cant be proven consistent

THUS A PARADOX

Godel must prove that a system cannot be proven to be consistent based upon the premise that the logic he uses must be consistent . If the logic he uses is not consistent then he cannot make a proof that is consistent. So he must assume that his logic is consistent so he can make a proof of the impossibility of proving a system to be consistent. But if his proof is true then he has proved that the logic he
uses to make the proof must be consistent, but his proof proves that
this cannot be done
THUS A PARADOX

Definition of LNC: To express the fact that the law is tenseless and to avoid equivocation, sometimes the law is amended to say "contradictory propositions cannot both be true 'at the same time and in the same sense'".

Formulation of LNC as a propositional statement: ( not ( A == B ) and (A == not B ) ) => True

P1. The Physical Universe is consistent
P2. If there exists any two objects in the PHYSICAL UNIVERSE such that the LNC returns False then the LNC itself is a contradiction.

a) “Any effectively generated theory capable of expressing elementary arithmetic cannot be both consistent and complete. In particular, for any consistent, effectively generated formal theory that proves certain basic arithmetic truths, there is an arithmetical statement that is true,[1] but not provable in the theory (Kleene 1967, p. 250)

note
"... there is an arithmetical statement that is true..."

In other words there are true mathematical statements which cant be proven
But the fact is Godel cant tell us what makes a mathematical statement true thus his theorem is meaningless

b) Gödel uses a G statement to prove his theorem but Godel's G statement is outlawed by the very axiom of the system he uses to prove his theorem -thus his proof is invalid,

Godel's G statement is impredicative '
Godel uses the axiom of reducibility -ie the logic of the system he is using to make his proof
but
the axiom of reducibility bans impredicative statements
thus Godel's G statement is banned-but he uses it to prove his theorem
thus Godel's theorem is invalid

2)Godel's 2nd theorem ends in paradox

Godel's 2nd theorem is about

"If an axiomatic system can be proven to be consistent and complete from
within itself, then it is inconsistent.”

But we have a paradox

Gödel is using a mathematical system
his theorem says a system cant be proven consistent

THUS A PARADOX

Godel must prove that a system cannot be proven to be consistent based upon the premise that the logic he uses must be consistent . If the logic he uses is not consistent then he cannot make a proof that is consistent. So he must assume that his logic is consistent so he can make a proof of the impossibility of proving a system to be consistent. But if his proof is true then he has proved that the logic he
uses to make the proof must be consistent, but his proof proves that
this cannot be done
THUS A PARADOX

Welcome to the discussion anne. You come from the same initial conclusion precisely as I have also and appreciate others who directly read and understood the topic.

Some initial recognition for others not in the know of the history, any discussions of this topic treated mathematics on par with logic and while some had different views of separation, everyone then treated the term, "arithmetic" as a particular subset of logic. Gödel's literal theorems were derived upon Russell's and Whitehead's attempt to prove everything from an initial set of simple primitives through a set of volumes of books under the title, "Principia Mathematica", for its main title in honor of Newton's same effort more normally treated for its physics that most ignore was more broad to include a universal means to prove everything from scratch.

Although this theorem is itself 'paradoxical', he, as well as those like Turing and Church, still showed that when we begin with a 'consistent and complete' logic (the 'first-order' logic Gödel used in his theorem), you can prove that it still leads permanently to a paradox, precisely as you concluded. He was trying to be very particular and had to in order to provide confidence of others who relied on the belief that formal logic itself was sufficient to resolve all problems in the realm of 'totality' (not simply our specific universe but to the total collection of all regardless of what one might believe).

Gödel knew this and was equally disappointed afterwards. His resolution was to adapt an 'intuitionist' stance that was more broad and related to a multi-valued system of reasoning that denied the "Law of Excluded Middle" (LEM by logik's shorthand here). This is where logik here is beginning from but why I thought it would be good to step back in this thread to look at its origins more specifically. You're among similar minds but with potential distinct approaches.

What should be noted to be more clear is that Gödel DID begin with a complete and consistent logic and so was't implying that that level of reasoning was not correct but that if you want to prove the paradox, you have to begin with the minimum premises that others are defaulted to thinking is true about logic, which included a need for binary values regarding 'truth'. Even before the "Principia", Russell wrote his first major book called, "Principles of Mathematics" (the same title but in English), as his first introduction to the problem. He intended THAT to be expanded upon until he realized certain concerns about his own approach and then restarted with Whitehead for the latter "Principia". He only tried to address the problem in the appendix, called "Types", to which he segregated the types of 'classes' that could or could not speak about themselves. (like a set defined as "the set of all sets" or "the sets of all sets not included in themselves").

Zermelo-Frankel Set theory opted to 'intuit' the axioms necessary to prevent this but it felt like a cheat or 'made up' because it had rules for creating new sets from old ones (the 'construction' concept which logik here may discuss as he has in separate threads.) Russel tried to be even more basic but defaulted to presuming the very underlying assumptions of 'consistency' (including 'non-contradiction' and 'exclusion of the middle' -- ie. exclusion of degrees of truth value).

Since 'degrees' of value in math exist as useful, if they are to mean anything for things like science, they too have to be proven with the same vigor and be inclusive from the ground-level logic that was intended to be discovered through the motivating questions, "can there be a universal logic;....and can that universal logic be extended to reality itself?" [a summary expression of my motive questions above.]

The concern begins in Calculus and the set theoretician, Cantor, who later discussed degrees of infinities which perplexed him. For instance, an "infinitesimal" is a distinct set of infinite Real numbers between any two integers. They are 'bounded' which makes them complete but hard to compare between other types of infinity. (like is there the same number of infinites between say, 0 to 1 as there is between 0 to 2?)

While is seems that we can begin in a 'quanta' like way (using definite integer wholes that we don't permit breaking down further than the whole), this 'discrete' approach cannot necessarily prove the 'continuities' that exist in all of reality. Nature appears to be contradictory this way. But some believed (hoped) this could be resolved and why the mathematician/scientist Hilbert formally announced a 'program' challenge for all logicians/scientists to try to address. This was to find some closed way of expressing all of reality without 'cheating' by requiring the introduction of some 'God of the gaps' to fill in the missing information.

Gödel's theories to me are better treated through the visualization that Turing opted to do that represented both 'logic' abstractly and realistically (because many then as now cannot conceive of math as being anything more than an invented but unreal tool.) That Gödel did so formally through logic, Turing added the feature of a physical factor that conforms to that logic which connects the logic to the practical representations of it physically.

Gödel specifically conditioned the argument with ,"effectively generated formal theory", precisely to indicate that his own logic is sufficiently complete and consistent. So he was meaning that you can't prove ALL systems of logic from only the 'consistent' assumption as a prerequisite for all other systems. The more inclusive reasoning of totality begins without ANY laws and why some, like myself, hold a type of "Logical Nihilism" extended to reality ignoring moral implications. [The label "Logical Nihilism" is used in moral philosophy but implies reality without any moral essence is equivalent to a reality without 'gods' that dictate consistent presumptions about 'fitness' as meaning something with value, like 'good' or 'healthy' or 'supreme' etc.]

So Gödel still presented a theorem that is 'valid' but that shows that it leads to some systems of logic that lacked 'consistency'. To give a simpler example regarding machines that 'function', ...we expect any machine to have a unique outcome to a specific set of inputs. If it has more than one conclusion, than this system is deemed a "relation" but not a function and so such a system would be like having a calculator that specifically FLIPPED between expected outcomes unpredictably.

The area of math that deals with this with respect to this indeterminate unique and predictable outcome, is "statistics". As such, while the statistics is math that can have initial assumptions that we know are valid in that system (consistent), it cannot be derived from a "first-order" logic without dealing with multiple 'universals' (or literal Universes in the case of physics). So it is to these kinds of 'arithmetics' (meaning any math systems) that you cannot start ONLY with a first-order CONSISTENT logic. You need one that extends itself with some assumptions that allow alternate outcomes (indetermine uniquely expected ones). These are the kinds of logics that are 'true' to us but not both consistent and complete.

The addition of the second, somewhat corollary about any system unable to prove itself, was to indicate precisely that if we find some universal system of "logic", it can't be proven as 'logical' because we expect it to be 'consistent' to some degree. Because a truly universal system of reasoning about nature cannot begin in a consistent manner [see my last note post above to logik at the end], you can have a 'rationale' of Totality that originates with NO LOGIC at all!

And that 'no logic' at all is rational at that stage is such precisely for NOT having even 'laws' to exist there in the extreme case of an absolute 'origin'. At that 'origin' where multiple realities can manifest themselves, where contradiction occurs, totality SPLITS those two options into discrete worlds that cannot be sensed from one another.

We may not have a logic that is consistent with nature. But we have many logics which are consistent with the human experience of nature.

And that's good enough. We care about updating the map, not the territory.

I'm concerned about the theoretical still because while it may appear trivial, the tendency for abuse is almost certain by those who capitalize on the institutions of 'wisdom', which is a result of the emphasis of 'practicality' alone.

Logic is still intrinsic to nature with priority or it will be reduced to an elite who maintains power by the dictating to others how 'practical' it is to obey their own laws of authority. That is, 'logic', if only treated as a made-up construct, makes those in power able to justify their own 'logic' arbitrarily. If nature is dependent upon 'laws of physics', it is dependent upon the concept of 'law' as a reality to be real. And this 'law', though not universal to our consistent ideals, has to itself have some 'law' of human behavior to prevent such skepticism needed to overrule those who maintain the power of their own 'laws' that enslave us.

We may not have a logic that is consistent with nature. But we have many logics which are consistent with the human experience of nature.

And that's good enough. We care about updating the map, not the territory.

I'm concerned about the theoretical still because while it may appear trivial, the tendency for abuse is almost certain by those who capitalize on the institutions of 'wisdom', which is a result of the emphasis of 'practicality' alone.

Logic is still intrinsic to nature with priority or it will be reduced to an elite who maintains power by the dictating to others how 'practical' it is to obey their own laws of authority. That is, 'logic', if only treated as a made-up construct, makes those in power able to justify their own 'logic' arbitrarily. If nature is dependent upon 'laws of physics', it is dependent upon the concept of 'law' as a reality to be real. And this 'law', though not universal to our consistent ideals, has to itself have some 'law' of human behavior to prevent such skepticism needed to overrule those who maintain the power of their own 'laws' that enslave us.

Logic is the assumption that nature has a structure. Logic attempts to mimic that structure.
It is a made-up construct. It's just language so how else should it be treated ?

What most logicians don't realize is that the structure of reality is many orders of magnitude more complex than any language we have.

Mistaking the complex for the simple...

The program which I used to construct the proof is 14000 CPU instructions which in turn depend on the 50 million transistors in the CPU.
When last did you ever evaluate logical expression with more than 20 ?

We may not have a logic that is consistent with nature. But we have many logics which are consistent with the human experience of nature.

And that's good enough. We care about updating the map, not the territory.

I'm concerned about the theoretical still because while it may appear trivial, the tendency for abuse is almost certain by those who capitalize on the institutions of 'wisdom', which is a result of the emphasis of 'practicality' alone.

Logic is still intrinsic to nature with priority or it will be reduced to an elite who maintains power by the dictating to others how 'practical' it is to obey their own laws of authority. That is, 'logic', if only treated as a made-up construct, makes those in power able to justify their own 'logic' arbitrarily. If nature is dependent upon 'laws of physics', it is dependent upon the concept of 'law' as a reality to be real. And this 'law', though not universal to our consistent ideals, has to itself have some 'law' of human behavior to prevent such skepticism needed to overrule those who maintain the power of their own 'laws' that enslave us.

Logic is the assumption that nature has a structure. Logic attempts to mimic that structure.

What most logicians don't realize is that the structure of reality is many orders of magnitude more complex than any logic we have.

Mistaking the complex for the simple...

The program which I used to construct the proof is 14000 lines of CPU instructions.

Logic, just as math too, acts to express the patterns in reality as 'laws', ...when 'consistent'. The function of logic, is science, but gets treated as its science's handmaiden.

Complex reality still has to be presumed originating from consistency and based on simple laws for our own world or it turns into religion. This is occurring even now with the Quantum Mechanics' Copenhagen Interpretation, Big Bang interpretation (versus, say, Steady State), and even the tendency to demand 'empirical-only' beliefs about what science as a whole is, are creating problems. This is because the means to which the theories that get 'officiated' are political in nature and prevent change when the institutions that manage them become more powerful and thus, authoritative.

I'm not sure what your last line was assuming? Complexity is not dismissive of its component simpler rules underneath it. "Complexity" refers to the quantity and multiple compounding of the simplicity at its roots. "Abstraction" is what is used to ignore that complexity of the simpler logic it is composed of, which is what you do when you operate top-down. It doesn't detract the significance of it but is about the 'practicality' of distinct separation of subject matters given we can't live forever to learn that complexity underneath. But it also has the problem in that those dependent upon learning logic FROM their abstracted perspective mistakes the complex for the simple, a direct opposite concern.

For example, those dependent upon cell phones by children growing up today may learn the abstract layers of reasoning about how to use the cell phone. They default to assuming the cell phone is no 'surprise' of existence for not being born in a time that there wasn't one. So they opt to presume that level is itself sufficient to trust without question unless some crisis comes along that compels them to deal with the underlying problems. That is, they mistaken their own 'logic' abstraction (their known use of cell/smart phones as real) as the foundation of 'simplicity'. So if one were to say challenge that as the legitimate layer of simplicity, they deny it for the sake they can 'empirically' see it work. But they become spoiled to the degree that they are dependent upon it without notice and dismiss any states where there is no such technology, like:

"Why do I need to learn math when my calculator can do the work for me?" [Spoiled upon the assumption that calculators are universally present and sufficiently infallible.]

"Why do I need to waste time on negative thinkers who complain how they can't get X, Y, and Z, when my own reality of being positive proves that I CAN get X, Y, and Z?" [Spoiled for interpreting what they get is due to their own power of 'positivity' when the opposite is more rationally true: that their capacity to get X, Y and Z with ease is precisely what makes them 'positive']

My point should be clear by these examples. The opposite is more the case: that people treat the complex for simplicity when they themselves function with the ease of accepting complexity and wonder as something not to be questioned because they don't need to question what they already get for not questioning. But this is the case for those most empowered by default. So the expectation upon those who actually require thinking are being imposed to shut up and trust those who are proven 'successful' for their power to be authoritative. It's a dumb-down effect that cycles through history creating fluctuations of intellectual and dark ages.

Godel 1st theorem is meaningless: as Godel cant tell us what makes a mathematics statement true-thus his theorem is meaningless
Godels 1st theorem is invalid as his G statement -which he uses to prove his theorem-is banned by an axiom of the system he uses to prove his theorem

Godel 1st theorem is meaningless: as Godel cant tell us what makes a mathematics statement true-thus his theorem is meaningless
Godels 1st theorem is invalid as his G statement -which he uses to prove his theorem-is banned by an axiom of the system he uses to prove his theorem

The closed or complete and consistent systems can solve an infinite set of problems but is 'bound' like the real numbers between any two integers (ie, infinitesimal). Thus a computer or logic system based on this can't solve ALL problems when expected to be 'finite', like the irrational numbers in a bound set. So the incompleteness theorems relate to the irrational number problem of the ancients. They wanted a mechanical process that could make all numbers represented as a ratio of integers (thus rational). The 'real' number system includes irrational numbers but cannot be listed finitely without a continuous process that never 'halts'.

What I can say is that I think that the various limit proofs don't require the depth expended in them when this can be more simply proven using the rational-irrational explanation intuitively. Another extended example is how a simple LINEAR design for a computer prior to multiple chips, could technically solve all problems but not in a finite time. The design to create parallel threads and cores require new distinct designing to provide being able to solve what the linear single processor couldn't do.

If anything, the limitations are resolvable by dimensioning, but this requires a 'constructive' type of logic with additional assumptions.You can have a simpler logic that begins with an Absolute Nothingness though. But most cannot fathom this idea and prefer an infinite version of origins. Then that cannot be closed either.

The essence of all the limit-related logic theorems or conjectures is about the same limits expected of science. If you don't think you can 'speak' of worlds beyond our own direct senses, then that limitation (sensing) prevents you from closure to all problems in Totality. This still can be proven as essential or we run into contradictions, like the Copenhagen interpretation for Quantum Mechanics.

Note, if you opt to read Turing's paper, his pre-requisite is that the system is "circle-free". The author of Turing Essentials didn't seem to understand what he meant correctly but it refers to dismissing repeat decimals as rationals, such as 1/3 = 0.33333.... To do this, he required the tape he used to not be connected in a literal circle which would force it to be 'finitely' limited.